A Proof of the Optimal Leapfrogging Conjecture

TL;DR

This paper proves the conjecture that the maximum speed of non-speed-of-light configurations in a generalized checkers game on lattices is 2/3 in 2D, extending previous work on maximum speed measures.

Contribution

It provides a proof of the optimal leapfrogging conjecture for the 2D case and introduces a framework for potential extension to higher dimensions.

Findings

Maximum speed of non-speed-of-light configurations is 2/3 in 2D.

Confirmed the conjecture for the 2D case.

Framework proposed for higher-dimensional generalizations.

Abstract

Suppose we place checkers in the lower left corner of a Go board and wish to move them to the upper right corner in as few moves as possible, where the pieces move as in the game of Chinese checkers. Auslander, Benjamin, and Wilkerson in 1993 generalized this game for integer lattices and defined a measure of speed for a starting configuration of pieces. They proved that the maximum speed of any configuration is 1, and only three configurations, called "speed-of-light" configurations, attain this speed. We prove their conjecture that the maximum speed of a non-speed-of-light configuration is 2/3 in the 2-dimensional case, and present a framework that should extend to higher dimensions.